Zhang Neural Network Model Research Articles

Gradient-Feedback Zhang Neural Network for Unconstrained Time-Variant Convex Optimization and Robot Manipulator Application

Optimization problems are frequently encountered in various fields. In this paper, the unconstrained time-variant convex optimization (UTVCO) problem is investigated. Generally, gradient neural network (GNN) is a traditional and effective method for solving time-invariant problems by making use of the gradient information. However, GNN is less effective on time-variant problems. On the other hand, Zhang neural network (ZNN) performs well on time-variant problems by exploiting the time-derivative information. In order to solve the UTVCO problem effectively and quickly with the help of the gradient information, inspired by the two methods, gradient-feedback ZNN (GZNN) is presented by taking both advantages of GNN and ZNN to solve the UTVCO problem. The main contributions are presented as follows. 1) The GZNN model for solving the UTVCO problem is proposed and analyzed theoretically. 2) Comparisons among GZNN, ZNN, GNN, and other models are presented with detailed discussions. Suggestions are provided on how to choose a model for solving the UTVCO problem better. 3) Tracking control of the robot manipulator is formulated as a UTVCO problem and studied with the GZNN model. According to the simulative and physical experiments, the task of tracking control is accomplished excellently by using the GZNN model.

Two New Zhang Neural Networks for Solving Time-Varying Linear Equations and Inequalities Systems.

Recently, Xu et al. solved a class of time-varying linear equations and inequalities systems (LEIESs) by using a Zhang neural network (ZNN) model through introducing a nonnegative relaxation vector. However, the introduction of this unknown nonnegative slack vector will increase the size and complexity of the model, thereby increasing the cost of computation. In this article, we propose two new ZNN models (called traditional Zhang neural network (TZNN) and variant Zhang neural network (VZNN) models, respectively) in which no additional relaxation vector is needed. The convergence analysis of these two new models are performed, and two simulation experiments are given to illustrate their efficiency and effectiveness for solving the time-varying LEIESs, including the applicability of our proposed models to robot manipulator.

General 9-instant discrete-time Zhang neural network for time-dependent applications

Zhang neural network (ZNN) is widely applied to solving time-dependent problems. For the sake of the implementation on the digital hardware platform, ZNN models need to be discretized. In this paper, as a further study of Zhang et al. discretization (ZeaD) formulas, a novel general 9-instant ZeaD formula is presented, and clear constraints are firstly given with proof. To evaluate the presented 9-instant ZeaD formula, three continuous-time models for time-dependent matrix inversion and pseudoinversion are presented with the help of Getz-Marsden dynamic system (GMDS) and ZNN. Then the corresponding discrete-time models are obtained by using the 9-instant ZeaD formula. According to the comparison experiments, the 9-instant ZeaD formula is substantiated to be effective and consistent with the theory. Furthermore, the problem of mobile angle-of-arrival (AoA) localization is investigated as a more specific and practical problem. In order to overcome the singularity problem of the tangent function in the representation of the AoA localization system, a new representation with sine and cosine functions is presented. Similarly, the continuous-time model is derived and discretized. Through comparison experiments, the discrete-time model obtained by the 9-instant ZeaD formula achieves desirable results, which further show the efficacy of the 9-instant ZeaD formula.

Discrete-time future nonlinear neural optimization with equality constraint based on ten-instant ZTD formula

In order to facilitate practical applications, discrete-time future nonlinear neural optimization with equality constraint (DFNNOEC) is investigated. First, starting from the continuous time-varying nonlinear neural optimization with equality constraint (CTNNOEC), continuous time-varying ZNN (CTZNN) models are derived by using Zhang neural network (ZNN) method. Then, a new ten-instant Zhang time discretization (TZTD) formula with high precision is presented to develop discrete algorithms for solving DFNNOEC problem in real time. Subsequently, two ten-instant discrete-time future ZNN (TDFZNN) algorithms are developed by utilizing TZTD formula to discretize CTZNN models. In addition, theoretical analyses expound the validity of TDFZNN algorithms. Finally, experiments of a numerical example and computer simulations on UR5 manipulator are carried out to confirm the effectiveness and superiority of TDFZNN algorithms. At the same time, computer simulation and physical experiment on Kinova Jaco2 manipulator further substantiate the practicability of TDFZNN algorithms.

Discrete-time ZNN-based noise-handling ten-instant algorithm solving Yang-Baxter-like matrix equation with disturbances

Time-variant Yang-Baxter-like matrix equation (YBLME) with the disturbances of noises is the hotspot in various scientific disciplines. The existing research, either can not handle the noise, or rely on the build-in numerical algorithm provided by MATLAB. Therefore, it is necessary to further study the digital-computer discrete solution of this problem. In this paper, the authors propose a noise-handling ten-instant discrete solution for the time-variant YBLME. By defining the matrix-form error function, Zhang neural network (ZNN) is exploited to design the continuous-time noise-handling ZNN model. For potential digital hardware (i.e., digital computer) realization, a discrete-time ZNN-based noise-handling ten-instant (DTZNH10i) algorithm, which is capable of handling three types of noises, is proposed on the basis of a Zhang time discretization (ZTD) formula. The convergence and precision of the proposed DTZNH10i algorithm are investigated and discussed. Theoretical analyses show the correctness and effectiveness of the proposed algorithm. The paper offers three examples of time-variant YBLME with disturbances of constant, linear, and bounded random noises to illustrate the efficacy and convergence of the proposed algorithm.

A novel hybrid Zhang neural network model for time-varying matrix inversion

A new hybrid Zhang neural network (HZNN) model is formulated to solve time-varying matrix inversion problem. The network is designed such that it encompasses the goodness of both gradient neural network (GNN) and Zhang neural network (ZNN). The stability of the network and convergence results are established theoretically. The rate of convergence of HZNN model is much faster compared to the classical GNN model and ZNN model for time-varying matrix inversion problem. Nonlinear activation functions for HZNN need not have to be odd and monotonically increasing for the network to converge, however with certain restrictions. Simulations are carried out to test the efficiency of HZNN over GNN and ZNN. Also, HZNN with odd and non-odd activation functions are examined for convergence. It is observed that the network converges much faster with the proposed non-odd function 3 (nof3). Furthermore, the HZNN model with odd and non-odd functions are applied to the kinematic control of a two-link planar manipulator for tracking a circular path.

A Vary-Parameter Convergence-Accelerated Recurrent Neural Network for Online Solving Dynamic Matrix Pseudoinverse and its Robot Application

Among this study, a vary-parameter convergence-accelerated neural network (VPCANN) model is generalized to solving dynamic matrix pseudoinverse, which can achieve super exponential convergence and noise-resistant, compared to the traditional Zhang neural network (ZNN) designed for dynamic problems. Simulative experiments reveal that the neural state solutions synthesized by the VPCANN can quickly approach to the theoretical pseudoinverse. Moreover, based on three types of noise disturbance including constant noise, random noise and dynamic noise, comparisons between the VPCANN and ZNN model are also investigated, verifying noise-resistant of the VPCANN model is better than the ZNN. In addition, to show the potential application of the VPCANN in practice, the kinematic motion planning of a six-links robot manipulator is considered, further substantiating the efficacy of the VPCANN in the dynamic matrix pseudoinverse.

Improved recurrent neural networks for solving Moore-Penrose inverse of real-time full-rank matrix

Recently, motivated by Zhang neural network (ZNN) models, Lv et al. presented two novel neural network (NNN) models for solving Moore-Penrose inverse of a time-invariant full-rank matrix. The NNN models were established by introducing two new matrix factors in the ZNN models, which results in their higher convergence rates than those of the ZNN models. In this paper we extend the NNN models to the more general cases through introducing a “regularization” parameter and a power parameter in these two matrix factors. The new proposed models are named here as the improved recurrent neural networks (IMRNN) since their convergence performance can be much better than the NNN models by appropriate choices of the introduced parameters. Such convergence property is theoretically analyzed in detail. Some numerical experiments are also performed to validate the theoretical results, including the numerical comparisons with the existing gradient neural network (GNN), ZNN and NNN models. In particular, the proposed IMRNN models are successfully applied to the inverse kinematic control of a three-link redundant robot manipulator where the superiority of the IMRNN models to the GNN, ZNN and NNN models is also indicated.

Li-Function Activated Zhang Neural Network for Online Solution of Time-Varying Linear Matrix Inequality

In the previous work, a typical recurrent neural network termed Zhang neural network (ZNN) has been developed for various time-varying problems solving. Based on the previous work, by exploiting a special activation function (i.e., Li activation function), the resultant ZNN model is presented and investigated in this paper for online solution of time-varying linear matrix inequality (TVLMI). For such a Li-function activated ZNN (LFAZNN) model, theoretical results are provided to show its excellent computational performance on solving the TVLMI. That is, the presented LFAZNN model has the property of finite-time convergence. Comparative simulation results with two illustrative examples further substantiate the efficacy of the presented LFAZNN model for TVLMI solving.

Computing Time-Varying ML-Weighted Pseudoinverse by the Zhang Neural Networks

The Zhang neural network (ZNN), a recurrent neural network, proposed in 2001, is particularly effective in solving time-varying problems. It has shown high efficiency and excellent performance in various applications. The weighted pseudoinverse is a useful tool in solving and analyzing the constrained least-squares problems. In this paper, we propose a ZNN model for computing the weighted pseudoinverse of a time-varying matrix. We show that our model converges globally and exponentially to the solution and our system is robust at the presence of small errors. A Matlab Simulink implementation of our model is presented. Our convergence analysis is verified by our experiments on testing matrices. A comparison study shows that our model has superior performance over the conventional gradient-based neural networks.

A novel noise-tolerant Zhang neural network for time-varying Lyapunov equation

The Zhang neural network (ZNN) has become a benchmark solver for various time-varying problems solving. In this paper, leveraging a novel design formula, a noise-tolerant continuous-time ZNN (NTCTZNN) model is deliberately developed and analyzed for a time-varying Lyapunov equation, which inherits the exponential convergence rate of the classical CTZNN in a noiseless environment. Theoretical results show that for a time-varying Lyapunov equation with constant noise or time-varying linear noise, the proposed NTCTZNN model is convergent, no matter how large the noise is. For a time-varying Lyapunov equation with quadratic noise, the proposed NTCTZNN model converges to a constant which is reciprocal to the design parameter. These results indicate that the proposed NTCTZNN model has a stronger anti-noise capability than the traditional CTZNN. Beyond that, for potential digital hardware realization, the discrete-time version of the NTCTZNN model (NTDTZNN) is proposed on the basis of the Euler forward difference. Lastly, the efficacy and accuracy of the proposed NTCTZNN and NTDTZNN models are illustrated by some numerical examples.

Discrete ZNN models of Adams-Bashforth (AB) type solving various future problems with motion control of mobile manipulator

Zhang neural network (ZNN), being a special type of recurrent neural network, has shown powerful abilities to solve a great variety of continuous time-varying problems. In order to solve future problems (or termed discrete time-varying problems), discrete ZNN (DZNN) models should be developed. In this paper, four kinds of future problems, i.e., future linear system (FLS), future division (FD), future quadratic minimization (FQM), and future equality-constrained quadratic programming (FECQP), are investigated. The DZNN models of Adams-Bashforth (AB) type are thus proposed for solving four kinds of future problems. Compared with conventional DZNN models of Euler type and Taylor type, the performances of DZNN models of AB type in terms of accuracy are better. Meanwhile, the numerical results substantiate the efficacy and superiority of DZNN models of AB type for solving future problems. Furthermore, the motion control of mobile robot manipulator is conducted to substantiate the efficacy of DZNN models of AB type for solving future problems.

Analysis and Application of Modified ZNN Design With Robustness Against Harmonic Noise

The Zhang neural network (ZNN) has recently realized remarkable success in solving time-varying problems. Harmonic noise widely exists in industrial applications and can severely affect the solution computed by ZNN models. This article attempts to solve the aforementioned limitations by providing the first ZNN design with an inherent capability to prohibit harmonic noise. Moreover, it opens new opportunities to shift the research on ZNNs in ideal situations to that with theoretical consideration on nonideal working environments. We establish a modified ZNN design formula in a noisy environment by incorporating the dynamics of harmonic signals. Theoretical analysis shows the convergence of the proposed ZNN design. An application case study for the new ZNN model verifies its effectiveness for time-varying matrix inversion in the presence of harmonic noise. The simulation further substantiates the effectiveness, superiority, and application prospects of the proposed ZNN design.

Higher-Order ZNN Dynamics

Several improvements of the Zhang neural network (ZNN) dynamics for solving the time-varying matrix inversion problem are presented. Introduced ZNN dynamical design is termed as ZNN models of the order p, $$p\ge 2$$, and it is based on the analogy between the proposed continuous-time dynamical systems and underlying discrete-time pth order hyperpower iterative methods for computing the constant matrix inverse. Such ZNN design is denoted by $$\hbox {ZNN}_H^p$$. Particularly, the $$\hbox {ZNN}_H^2$$ design coincides with the standard ZNN design. Moreover, $$\hbox {ZNN}_H^3$$ design represents a time-varying generalization of the previously defined ZNNCM model. In addition, an integration-enhanced noise-handling $$\hbox {ZNN}_H^p$$ model, termed as $$\hbox {IENHZNN}_H^p$$, is introduced. In the time-invariant case, we present a hybrid enhancement of the $$\hbox {ZNN}_H^p$$ model, shortly termed as $$\hbox {HZNN}_H^p$$, and investigate it theoretically and numerically. Theoretical and numerical comparisons between the improved and standard ZNN dynamics are considered.

Noise-Tolerant and Finite-Time Convergent ZNN Models for Dynamic Matrix Moore–Penrose Inversion

Dynamic (or say, time-varying) problems have been a hot spot of research recently. As a general form of matrix inverse, dynamic Moore–Penrose inverse solving has received more and more attention owing to its broad applications. The approaches based on neural networks have become a popular solution to various dynamic matrix-related problems including dynamic Moore–Penrose inverse. However, existing neural models either only achieve infinite-time instead of finite-time convergence, or are sensitive to noises. Therefore, finite-time convergent neural model, which is simultaneously capable of addressing the noises, is desperately needed for dynamic Moore–Penrose inverse solving. To do that, in this paper, a novel evolution formula is designed based on the widely investigated Zhang neural network (ZNN). Accordingly, two modified ZNN models (MZNN), namely MZNN-R and MZNN-L models, are proposed and analyzed for the right and left dynamic Moore–Penrose inversion of full-rank matrices, respectively. In addition to providing detailed theoretical analyses on the desired finite-time convergence and noise-depression properties of the proposed two models, we also perform two numerical examples for further verification. Furthermore, to illustrate the potential of MZNN models in practical applications, two path-tracking control examples are also presented via a two-dimensional planar three-link and a three-dimensional Kinova Jaco $^2$ redundant robot manipulator. The feasibility, extraordinary efficacy, and superiority of the proposed MZNN models for dynamic Moore–Penrose inverse solving are corroborated by both theoretical results and simulation observations.

Discrete-Time Zhang Neural Networks for Time-Varying Nonlinear Optimization

As a special kind of recurrent neural networks, Zhang neural network (ZNN) has been successfully applied to various time-variant problems solving. In this paper, we present three Zhang et al. discretization (ZeaD) formulas, including a special two-step ZeaD formula, a general two-step ZeaD formula, and a general five-step ZeaD formula, and prove that the special and general two-step ZeaD formulas are convergent while the general five-step ZeaD formula is not zero-stable and thus is divergent. Then, to solve the time-varying nonlinear optimization (TVNO) in real time, based on the Taylor series expansion and the above two convergent two-step ZeaD formulas, we discrete the continuous-time ZNN (CTZNN) model of TVNO and thus get a special two-step discrete-time ZNN (DTZNN) model and a general two-step DTZNN model. Theoretical analyses indicate that the sequence generated by the first DTZNN model is divergent, while the sequence generated by the second DTZNN model is convergent. Furthermore, for the step-size of the second DTZNN model, its tight upper bound and the optimal step-size are also discussed. Finally, some numerical results and comparisons are provided and analyzed to substantiate the efficacy of the proposed DTZNN models.

A Finite‐Time Recurrent Neural Network for Computing Quadratic Minimization with Time‐Varying Coefficients

This paper proposes a Finite-time Zhang neural network (FTZNN) to solve time-varying quadratic minimization problems. Different from the original Zhang neural network (ZNN) that is specially designed to solve time-varying problems and possesses an exponential convergence property, the proposed neural network exploits a sign-bi-power activation function so that it can achieve the finite-time convergence. In addition, the upper bound of the finite convergence time for the FTZNN model is analytically estimated in theory. For comparative purposes, the original ZNN model is also presented to solve time-varying quadratic minimization problems. Numerical experiments are performed to evaluate and compare the performance of the original ZNN model and the FTZNN model. The results demonstrate that the FTZNN model is a more effective solution model for solving time-varying quadratic minimization problems.

Robustness Analysis and Robotic Application of Combined Function Activated RNN for Time-Varying Matrix Pseudo Inversion

As a typical representative of the recurrent neural network (RNN), Zhang neural network (ZNN) has been proved as a powerful parallel-processing neural solver for time-varying matrix problems. Recent studies have shown that, in the absence of noise, the ZNN model activated by a combined activation function (CAF), which is the linear combination of sign-bi-power (SBP) and linear functions (termed CAF-ZNN), can achieve much better finite-time convergence compared with other ZNN models for time-varying matrix pseudoinversion. This paper investigates for the first time the influence of noises on such a CAF-ZNN model. We not only present the upper bound of steady-state solution error synthesized by the noise-polluted CAF-ZNN model for time-varying matrix pseudoinversion but also derive the finite convergence time for the solution error reaching the upper bound. Both the theoretical analyses and simulation observation reveal that the solution error can be made arbitrarily small by choosing some design parameters of CAF-ZNN. At last, a robotic application is successfully performed to further illustrate the feasibility of the CAF-ZNN model to kinematic control of redundant robot manipulator.

Design and Analysis of Two FTRNN Models With Application to Time-Varying Sylvester Equation

In this paper, to accelerate the convergence speed of Zhang neural network (ZNN), two finite-time recurrent neural networks (FTRNNs) are presented via devising two novel design formulas. For verifying the advantages of the proposed FTRNN models, a solution application to time-varying Sylvester equation (TVSE) is given. Compared with the conventional ZNN model, the presented new FTRNN models in this paper are theoretically proved to have better convergence performance, and they are more effective for online solving TVSE within finite time. At last, the superiority and effectiveness of the new FTRNN models for solving TVSE are verified by numerical simulations.

A novel recurrent neural network and its finite-time solution to time-varying complex matrix inversion

A complex-valued nonlinear recurrent neural network is designed and researched for time-varying matrix inversion solving in complex field. Unlike the design methods of the conventional gradient neural network (CGNN) and the previous Zhang neural network (ZNN), the proposed complex-valued nonlinear recurrent neural network (CVNRNN) model is established on basis of a nonlinear evolution formula and possesses a better finite-time convergence Besides, the detailed theoretical analysis provides a guarantee for the finite-time convergence achievement of the CVNRNN model. In addition, the theoretical analysis is also verified by numerical simulations, which comparatively show that the proposed CVNRNN model is faster and more accurate than the ZNN model and the CGNN model in solving time-varying complex matrix inversion.